He also offered to ask parliamentary questions and was offered a monthly payment of £10,000 but said he could do a deal for £12,000 a month.

He told undercover reporters he offered “value for money” because he could introduce them to senior members of all three of the main political parties.

But in a statement sent to the Sunday Times, Lord Cunningham denied breaking any rules.

He said: “I deny any agreement to operate in breach of the House of Lords code of conduct and, in fact, recall that I made it clear that I would only operate within the rules.”

He said his reference to “a fanciful £12,000 a month payment” was made to test his suspicion that he had been talking to journalists.

Lord (Brian) Mackenzie, a former chief superintendent for Durham police and President of the Police Superintendents Association, said he could arrange parties for paying clients – including on the terrace of the House of Lords – after being asked if this was possible.

“I just say to a colleague who has nothing to do with it, ‘would you host a function for me?'” he said.

When asked if he had done anything wrong, Lord Mackenzie told BBC Radio 5 live: “Not at all”.

“There is nothing in the rules to prevent a peer hosting a function, as long as he has no financial interest. I was being interviewed in connection, I thought, with a position as a consultant for this energy company… not as a lobbyist.”

He said that the “complex” rules of lobbying “puts peers in a very difficult position”. “I have asked no questions for money,” he added.

The third peer, Lord Laird, said he could arrange to get other peers involved.

He explained that, working together, they could ask parliamentary questions for each other’s clients, put down amendments in debates or write to ministers.

“Some of the guys for instance in the Lords … I will get to put down questions for me … and then I put down questions for them,” he said.

But later, in a statement to the BBC, Lord Laird said he had been “the subject of a scam” by journalists.

“This has led to allegations that I have broken the rules of the House of Lords,” he said.

“I wish to make it clear that I did not agree to act as a paid advocate in any proceedings of the House nor did I accept payment or other incentive or reward in return for providing parliamentary advice or services.”

He said that, although he had not broken any rules, he had referred himself “to the appropriate authorities” and would be making “no further statement until I have received their ruling”.

On Sunday morning, he announced that he had resigned the UUP whip in the House of Lords.

The Sunday Times said all three had told their undercover reporters at the time that they would declare any payment in the House of Lords register.

Labour said in a statement it expected all peers to comply with the Lords code of conduct and that, where there was evidence of wrongdoing, the party would consider disciplinary action.

In separate investigation by the Daily Telegraph in conjunction with BBC Panorama, Lord Laird told an undercover journalist: “If I take this on, I would be doing that for monetary reward and I don’t want to be having to unveil that because it weakens the position”

Lord Laird was approached by a fake lobbying company – claiming to represent business interests in Fiji.

He was secretly filmed being asked if he would be interested in being paid to work for the company.

He said he would arrange for questions to be raised in Parliament for a retainer of £2,000 a month.

The Ulster Unionist Party said Lord Laird contacted the party on Friday regarding the allegations.

A spokesman for the party said: “He assured the party that he has done nothing wrong, but in the interest of transparency, he has referred himself to the appropriate authorities at Westminster so that they can take a view on his conduct.

“The Ulster Unionist Party considers that this is the correct course of action.”

On Saturday, the BBC’s Panorama programme released footage – secretly filmed as part of the same investigation – of MP Patrick Mercer appearing to offer a Commons security pass to a fake Fijian firm that paid him £4,000 to ask parliamentary questions.

It was released a day after he resigned the Tory whip over the programme’s allegations that he broke Commons lobbying rules.

Mr Mercer said he took the money for consultancy work outside Parliament, that he was taking legal advice and that he had referred himself to Parliament’s standards commissioner.

On Thursday, Panorama will air allegations from its joint investigation with the Daily Telegraph.